To the end of this section, for easiness of reading let us drop the bar over new variables. Now, following the general idea of the method of heat potentials for pricing double barrier options, [1]}, [2]}, we represent each function \(q_i(\tau ,x)\)  as
 
qi(i,x) = 0i { i(k) G(x, , i - k)|= yi(k) + i(k) G(x,, i - k)|= yi+1(k) }dk.

In () the second integral is a sum of two single layer potentials with the potential densities \(\Psi _i(\tau )\)  and \(\Phi _i(\tau )\) . By writing () we take into account that according to (REF ) and (REF ), the transformed time \(\tau \)  might differ for each interval, therefore, the notation \(\tau _i\)  is used. However, e.g., for the problem described in Section REF  all new times are equal, i.e. \(\tau _i = \tau , \ i=1,\ldots ,N+1\) .
